Subjective Morality?

[DLJ]

OK.  So, I'm beginning to understand that you see this as 'sides'.  And that 'moral facts' are dogma that one can only accept once one has subconsciously chosen a side (deduced from post #263).  Is there a group name for these sides (e.g. sect, cult, religion, tribe, team, position, school, whatever)?

They're called meta ethical theories or systems.  Trying to get at the ontology of morality.  As opposed to applied or normative ethical theories, or moral epistemology.  
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Thank you. 'Systems' is my preference; I'll use that.

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And what is the tribal name (team name / side etc.) for one who holds that morality is both cognitive and non-cognitive?

Irrational and/or confused would be that tribes name.  The positions describe baseward exclusions. 
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I have no idea what 'baseward' means.

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If you think that our moral propositions express a state of belief,
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My question was about 'morality' not 'moral propositions'.

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That person would be negotiating over how much of morality as practiced is properly non cognitive, not objecting to the fact that our moral propositions can or do express states of belief.

Exactly.